Wednesday, January 7, 2015

Alternating-Current Circuits Containing Resistance : Current and voltage in phase , Heating effect of an alternating current , Effective (RMS) voltage and Resistance

Alternating-Current Circuits Containing Resistance

A simple alternating-current circuit consists of resistance only. Either an incandescent lighting load or a heating load, such as a heater element, is a noninductive resistive load.

Operation of an AC Generator

In Figure 2–1, an ac generator supplies current to a 100-ohm (D) heater element. The output of this generator is a voltage sine wave with a maximum or peak value of 141.4 V. When the voltage in this circuit is zero, the current is zero. When the voltage is


at its maximum value, the current is also at maximum. When the voltage reverses direction, the current also reverses direction. When the current and the voltage waveforms of a circuit are zero at the same time and reach their maximum values at the same time and in the same direction, these waves are said to be in phase.

Figure 2–2 shows the voltage and current sine waves in phase for the circuit of Figure 2–1. Ohm’s law states that the current in a resistor is directly proportional to the voltage and inversely proportional to the magnitude of the resistance of the circuit. Note in Figure 2–2 that as the voltage increases from zero in either direction, the current increases proportionally in the same direction as required by Ohm’s law. Thus, Ohm’s law may be applied to ac circuits having a resistive load:



The alternating current in the circuit of Figure 2–1 is shown by the sine wave in Figure 2–3. The instantaneous maximum value of the current is 1.414 A.

If the current wave is considered for one complete cycle, then its average value is zero. This value is due to the fact that the negative alternation is equal to the positive alternation. If a dc ammeter is used to measure the current of this circuit, it will indicate zero. Thus, alternating current must be measured using an ac ammeter. Such an instrument measures the effective value of the current.

The effective value of alternating current is based on its heating effect and not on the average value of a sine-wave pattern. An alternating current with an effective value of one ampere is that current that will produce heat in a given resistance at the same rate as one ampere of direct current.

Plotting Sine Waves of Current

Direct Current Fundamentals showed that the heating effect of ac varies as the square of the current (watts I2R). For alternating current with a maximum value of 1.414 A, it is possible to plot a curve of the squared values of the current. If the horizontal scale of the curve is graduated in electrical time degrees, it is a relatively simple matter to obtain the instantaneous current values at regular intervals, such as 15° increments. The instantaneous values of current can then be squared and plotted to give a curve of current squared values for one cycle. For example, the instantaneous value of current at 30° is determined as follows:



direction of current is reversed in the negative alternation of the cycle. The current squared values are all positive. (Recall that when any two negative numbers are multiplied, the product is positive.)

Graph of Sine Waves of Current. Figure 2–4 shows a sine wave of current that was plotted using the instantaneous current values given in Table 2–1. In addition, a current squared wave is also shown. This wave was plotted using the squared current values from Table 2–1.

The current squared wave in Figure 2–4 has a minimum value of 1.4142 or 2 A. Note that the entire current squared wave is positive because the square of a negative value is positive. The graph shows that the current squared wave has a frequency that is twice that of the sine wave of current. The average value of the current squared wave is 1.0 A and is indicated by the dashed line on the graph.

The two areas of the current squared wave above the dashed line are the same as the areas of the two shaded valleys below the dashed line. This average value of one ampere over a period of one cycle gives the same heating effect as one dc ampere. The rectangular area formed by the dashed line and the zero reference line represents the heating effect of this alternating current. It also shows the heating effect of one direct-current ampere over a period of one cycle.

Root-Mean-Square Value of AC Current

An alternating current with a maximum value of 1.414 A and a steady direct cur- rent of one ampere both produce the same average heating through a resistance in a period of one cycle. In other words, both currents have the same average squared value. This value is called the effective value or the root-mean-square (RMS) value of current. Root-mean-square current is the abbreviated form of “the square root of the mean of


the square of the instantaneous currents.” The RMS value of current is the current indicated by the typical ac ammeter. The relationship between the effective (RMS) current and the maximum current is


This means that the typical ac ammeter indicates 0.707 of the maximum value of a sine wave of current. This relationship can also be expressed as a ratio between the maxi- mum value and the RMS value, or M2, which is 1.414. For example, if an alternating current has an instantaneous maximum value of 20 A, an ac ammeter will read


Either of these two ratios can be used to find the maximum value of current when the RMS or effective value is known, For example, an ac ammeter indicates a value of 15 A. The maximum current is


In ac calculations, generally the effective value of current is used. This current is indicated by the letter I, which stands for intensity of current. The maximum current is shown as I .


If one RMS ampere of alternating current passes through a resistance of one ohm, the RMS voltage drop across the resistor is one RMS ac volt. The RMS ac volt is 0.707 of the instantaneous maximum voltage. This RMS voltage is called the effective voltage. The typical ac voltmeter reads the effective value of voltage. The relationship between the maximum voltage and the effective voltage is the same as the one between the maximum current and the effective current.

The effective value of voltage is generally used in ac calculations. The effective voltage is indicated by the letter V. The maximum voltage is usually indicated by V .

For example, an ac voltmeter is connected across a lighting circuit and indicates a value of 120 V. What is the maximum instantaneous voltage across the circuit?


Unless otherwise specified, the voltage and current in an alternating-current circuit are always given as effective values. All standard ac ammeters and voltmeters indicate effective or RMS values.

Defining Terms

The following terms all relate to the sine wave: average, instantaneous, effective or RMS, peak, maximum, and peak-to-peak. These terms tend to confuse the student because they are closely related to each other. Figure 2–5 illustrates the various terms and their relationships.


In alternating-current circuits, the resistance is due to incandescent lighting loads and heating loads, just as in dc circuits. For these loads, inductance, hysteresis effects, and eddy current effects may be neglected. In a later unit of this text, the discussion will cover those factors that can change the ac impedance (resistance) of various types of loads. The term


impedance is generally used to describe the total current limiting effect in alternating current circuits. Impedance is a combination of all current limiting properties such as resistance, inductance, and capacitance. Impedance will be discussed in later chapters of this text.